Given two polygonal curves $P$ and $Q$ defined by $n$ and $m$ vertices with $m\leq n$, we show that the discrete Fréchet distance in 1D cannot be approximated within a factor of $2-\varepsilon$ in $\mathcal{O}((nm)^{1-δ})$ time for any $\varepsilon, δ>0$ unless OVH fails. Using a similar construction, we extend this bound for curves in 2D under the continuous or discrete Fréchet distance and increase the approximation factor to $1+\sqrt{2}-\varepsilon$ (resp. $3-\varepsilon$) if the curves lie in the Euclidean space (resp. in the $L_\infty$-space). This strengthens the lower bound by Buchin, Ophelders, and Speckmann to the case where $m=n^α$ for $α\in(0,1)$ and increases the approximation factor of $1.001$ by Bringmann. For the discrete Fréchet distance in 1D, we provide an approximation algorithm with optimal approximation factor and almost optimal running time. Further, for curves in any dimension embedded in any $L_p$ space, we present a $(3+\varepsilon)$-approximation algorithm for the continuous and discrete Fréchet distance using $\mathcal{O}((n+m^2)\log n)$ time, which almost matches the approximation factor of the lower bound for the $L_\infty$ metric.

翻译：给定由$n$个和$m$个顶点定义的两条多边形曲线$P$和$Q$，其中$m\leq n$，我们证明：除非OVH（正交向量假设）不成立，否则一维离散弗雷歇距离无法在$\mathcal{O}((nm)^{1-δ})$时间内以$2-\varepsilon$的近似因子进行逼近，其中$\varepsilon, δ>0$。通过类似构造，我们将此下界推广到二维空间中连续或离散弗雷歇距离的情形，并将近似因子提升至$1+\sqrt{2}-\varepsilon$（对应欧几里得空间）或$3-\varepsilon$（对应$L_\infty$空间）。这强化了Buchin、Ophelders和Speckmann的下界结果，将其推广至$m=n^α$（$α\in(0,1)$）的情形，并将Bringmann提出的$1.001$近似因子进一步增大。针对一维离散弗雷歇距离，我们提出了一种具有最优近似因子和近乎最优运行时间的近似算法。此外，对于嵌入任意$L_p$空间的任意维度曲线，我们提出了一种$(3+\varepsilon)$-近似算法，用于计算连续和离散弗雷歇距离，其时间复杂度为$\mathcal{O}((n+m^2)\log n)$，该近似因子几乎达到了$L_\infty$度量下理论下界的对应值。